The Prescribed-Vertex Semidegree Threshold for Directed $3q$-Cycles in Oriented Graphs

Abstract: For every $q\ge2$, we prove that every oriented graph $G$ on $n\ge45q-8$ vertices whose minimum semidegree satisfies [ δ^0(G)\ge \left\lceil\frac n3\right\rceil ] contains a directed cycle of length $3q$ through every vertex. The semidegree bound is sharp. This closes the one-unit gap left by the prescribed-vertex theorem of Kelly, Kühn and Osthus when $3\mid n$. We also prove that if an oriented graph $H$ has order $N$, minimum semidegree $d\ge3$, and $7d\ge2N+3$, then every ordered pair of distinct vertices is joined by a path of length three, four, or five. The constant $+3$ is best possible. As a consequence, the order hypothesis $n\ge10<sup>{10}\ell$ in the general prescribed-vertex theorem of Kelly, Kühn and Osthus can be replaced by $n\ge15\ell-60$ for $\ell\ge7$.

• The paper determines the exact minimum semidegree threshold for directed cycles of length 3q in oriented graphs
• The threshold is exact for large order vertex oriented graphs with minimum semidegree roughly equal to one-third of the number of vertices and for all cycles where q >=2
• The prescribed-vertex threshold closed a gap in previous literature, proving $ au^v_{3q}(n)= ⌈ n/3 ⌉$ for all larger n, and replacing http://arxiv.org/abs/2608.20048 $10^{10}$

Overview and main result

This paper by Zhenhua Lyu determines exactly the minimum semidegree threshold that forces a directed cycle of length $3q$ through every prescribed vertex of an oriented graph. For $q\ge2$ and $n\ge n_0(q)$, where $n_0(q)=37$ for $q=2$, $97$ for $q=3$, and $45q-8$ for $q\ge4$, the paper proves that every oriented graph on $n$ vertices with $q\ge2$0 contains a copy of $q\ge2$1 through every vertex (2608.20048). The bound is sharp: a modified cyclic blow-up construction with $q\ge2$2 contains a vertex lying on no directed cycle whose length is divisible by three. Consequently, the prescribed-vertex threshold satisfies $q\ge2$3 for all $q\ge2$4.

The significance lies in closing the one-unit gap left by Kelly, Kühn and Osthus (KKO). Their theorem gives the threshold $q\ge2$5 under an enormous order hypothesis $q\ge2$6, while their cyclic blow-up lower bound shows $q\ge2$7. When $q\ge2$8 these bounds coincide, but when $q\ge2$9 they leave the window $n\ge n_0(q)$0 open. This paper resolves the critical case $n\ge n_0(q)$1 in favor of the lower bound, and simultaneously replaces the order hypothesis $n\ge n_0(q)$2 by an explicit linear one.

A sharp short-linking lemma

The technical core enabling the improved order bounds is a linking lemma: if an oriented graph $n\ge n_0(q)$3 of order $n\ge n_0(q)$4 has minimum semidegree $n\ge n_0(q)$5 and $n\ge n_0(q)$6, then every ordered pair of distinct vertices is joined by a directed path of length three, four, or five. The proof assumes no such path exists, builds a layered forbidden-arc structure from equal-sized sets inside $n\ge n_0(q)$7 and $n\ge n_0(q)$8, derives degree-counting inequalities, and reduces the contradiction to a convex quadratic whose two endpoint estimates use precisely the relation $n\ge n_0(q)$9, where $n_0(q)=37$0.

The additive constant is best possible: for every $n_0(q)=37$1 there is a graph of order $n_0(q)=37$2 with $n_0(q)=37$3 (so $n_0(q)=37$4) in which two specified vertices are joined only by paths of length one, two, or at least six. In defect form, $n_0(q)=37$5 guarantees short paths already when $n_0(q)=37$6, and the coefficient $n_0(q)=37$7 is asymptotically best possible—no constant $n_0(q)=37$8 yields a universal sufficient order hypothesis.

Two corollaries follow directly. First, taking $n_0(q)=37$9 appropriately recovers deletion-tolerant versions used throughout the main argument: after deleting at most $q=2$0 vertices from a graph with $q=2$1 and $q=2$2, all ordered pairs remain joined by short paths. Second, combining the lemma with KKO's butterfly construction improves their general prescribed-vertex theorem: for every $q=2$3 and $q=2$4, the condition $q=2$5 forces $q=2$6 through every vertex, replacing $q=2$7 by a linear bound while retaining both hypotheses. Together with KKO's direct treatments of $q=2$8, this yields explicit linear order bounds for all cycle lengths.

Structure of the equality case

Only $q=2$9 requires new arguments beyond the KKO framework. Fixing a vertex $97$0, the proof branches on whether outneighbourhoods are independent:

Butterfly branch: if neither $97$1 nor $97$2 (for an arc $97$3 in $97$4) is independent, the arcs $97$5 form an $97$6-butterfly containing $97$7–$97$8 paths of lengths two, three, and four. A cardinality argument shows that a butterfly plus $97$9 already forces $q=3$0 onto a $q=3$1: assuming no $q=3$2 through $q=3$3 forbids return paths from $q=3$4 to $q=3$5 of certain lengths, forcing four layers of neighbourhoods to be nearly disjoint, which contradicts $q=3$6 since their total size must then exceed $q=3$7; the forced overlap produces a forbidden length-four path avoiding $q=3$8. For $q=3$9, the butterfly's three path lengths combine with the short-linking lemma to close cycles of length $45q-8$0.

When $45q-8$1, every outneighbourhood has size $45q-8$2 exceeding the maximum independent set size $45q-8$3, so no outneighbourhood is independent and only the butterfly branch is needed—this explains why the equality analysis is confined to $45q-8$4.

Balanced cuts: if some relevant outneighbourhood is independent, Lemma on balanced cuts shows it has size exactly $45q-8$5, its complement has size $45q-8$6, and each vertex of the independent part sends and receives exactly $45q-8$7 arcs across the cut. The proof then splits into a root-dominating cut ($45q-8$8) and a transitive-entry cut (rooted at $45q-8$9), handled respectively by matching arguments over row families and by combinations of initial paths with short linking.

Row-family stability and matching

In a balanced cut, each $q\ge4$0 has an $q\ge4$1-element "row" $q\ge4$2. A key stability result classifies row families admitting no terminal-safe three-chain—a configuration yielding initial paths of lengths three, four, and five that short linking closes to $q\ge4$3. Via a rainbow-representative lemma for four equicardinal sets (proved by Hall's theorem with a careful analysis of same-singleton obstructions), absence of a terminal-safe chain forces either a single dominant row type or rows varying on at most three points.

Both stable outcomes yield dense bipartite graphs between the common intersection and its complement, and König's theorem supplies matchings of size $q\ge4$4 or $q\ge4$5: a hypothetical smaller matching would give a small vertex cover meeting too few edges against the density lower bound, contradicting the numerical conditions $q\ge4$6 (or weaker variants). Splicing the matching into alternating $q\ge4$7 segments around the root produces $q\ge4$8 explicitly.

A nonextendability lemma limits to three the number of row labels lacking a length-three transition, keeping the active row family large enough for these counting arguments. The three cutoffs in $q\ge4$9 trace to different deletion budgets in the terminal-safe-chain branches: $n$0 for $n$1, $n$2 for $n$3, and $n$4 for $n$5, each contributing $n$6.

Limitations and open questions

The paper concedes several boundaries. The restriction $n$7 is essential: the directed triangle threshold has asymptotic scale $n$8 rather than $n$9, so $q\ge2$00 falls outside this phenomenon. For $q\ge2$01, any argument using the residual graph only through its order and semidegree cannot lower $q\ge2$02: at $q\ge2$03 the terminal-safe-chain branch leaves parameters matching the sharp counterexample of the linking lemma, so further improvement requires exploiting additional structure of the deleted subgraph, such as the relation between indegree and outdegree losses. It remains open to determine the smallest constant $q\ge2$04 for which an order hypothesis $q\ge2$05 suffices in the main theorem, and similarly whether the coefficient in $q\ge2$06 is optimal. For $q\ge2$07 with $q\ge2$08, the stronger order bound $q\ge2$09 applies via the general corollary, but the equality-case thresholds for $q\ge2$10 rely on ad hoc budgets rather than a unified optimization.

Conclusion

The paper settles the prescribed-vertex semidegree threshold for directed $q\ge2$11-cycles at $q\ge2$12 for $q\ge2$13, resolving the residue-class-$q\ge2$14 ambiguity in the KKO window and replacing their astronomical order hypothesis with explicit linear bounds. The sharp short-linking lemma ($q\ge2$15) is independently valuable and drives both the equality analysis and the linearization of the general theorem. The remaining quantitative questions concern optimal linear constants rather than the qualitative threshold, which is now exact.